Analytical solution to the radiotherapy fractionation problem including dose bound constraints

Abstract

This paper deals with the classic radiotherapy dose fractionation problem for cancer tumors concerning the following goals: (a) To maximize the effect of radiation on the tumor, restricting the effect produced to an organ at risk (healing approach). (b) To minimize the effect of radiation on one organ at risk, while maintaining enough effect of radiation on the tumor (palliative approach). We will assume the linear-quadratic model to characterize the radiation effect without considering the tumor repopulation between doses. The main novelty with respect to previous works concerns the presence of minimum and maximum dose fractions, to achieve the minimum effect and to avoid undesirable side effects, respectively. We have characterized in which situations is more convenient the hypofractionated protocol (deliver few fractions with high dose per fraction) and in which ones the hyperfractionated regimen (deliver a large number of lower doses of radiation) is the optimal strategy. In all cases, analytical solutions to the problem are obtained in terms of the data.